Guided matter-wave sagnac interferometer

ABSTRACT

The present invention provides an interferometer apparatus comprising a matter-wave guide enclosing an area, wherein a flux of particles may be guided in the matter-wave guide in at least two opposite paths, the matter-wave guide is rotatable relative to an inertial frame of reference; a first beam splitter to split the first beam to at least second and third beams, each of the second and third beams is to be guided in another path of the two opposite paths; and a second beam splitter allowing particles of the second and third beams to exit the matter-wave guide in a first probability and to stay in the matter-wave guide in a second probability.

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. provisional application Ser. No. 60/877,591, which is incorporated in its entirety herein by reference.

BACKGROUND OF THE INVENTION

Sagnac effect is a phase shift induced in a wave propagating in a loop by rotation of the loop. Two waves propagating in opposite directions along a rotating closed loop may interfere in a point of exit, wherein the rotation may cause a phase shift between them. An optical Sagnac Interferometer apparatus may include, for example, a loop trajectory enclosing an area. The interferometer may operate by splitting a beam of light at an entrance of the loop, to two beams which may be made to propagate in opposite directions along the loop trajectory. An interference pattern may be obtained at a point of exit from the loop. When the apparatus rotates, the path length of one of the beams may be effectively shortened, while the path length of the other beam may be effectively lengthened. Therefore, a phase shift may be created between the two beams, dependent on the rotation velocity of the apparatus.

Therefore, a Sagnac interferometer may measure changes in the rotation frequency of the frame that it is fixed to, relative to a global (inertial) frame of reference. The Sagnac effect in a wave propagating through a closed rotating ring induces a phase shift proportional to the rotation frequency Ω of this rotation and the area A of the ring. For light waves with frequency ω this phase shift may be represented by φ_(light)=(2ωA/c²)Ω, where c is the speed of light and w is the light frequency. For matter waves of massive particles with mass m, the induced phase shift may be represented by φ_(matter)=(2mA/)Ω, which is larger than the phase shift in an optical SI having the same area A by mc²/ω, wherein  is planck's constant. Therefore, the Sagnac phase is much more sensitive to rotations in a matter-wave SI compared to light wave SI, for example, by 10 orders of magnitude. However, state-of-the-art matter-wave Sagnac interferometers, which are based on beams of atoms traveling in free space, are limited by their area number of atoms and the allowed momentum bandwidth of the atoms.

BRIEF DESCRIPTION OF THE DRAWINGS

The subject matter regarded as the invention is particularly pointed out and distinctly claimed in the concluding portion of the specification. The invention, however, both as to organization and method of operation, together with objects, features, and advantages thereof, may best be understood by reference to the following detailed description when read with the accompanying drawings in which:

FIG. 1 is a schematic illustration of a SI (Sagnac interferometer) according to some embodiments of the present invention;

FIG. 2 is an exemplary illustration of a SI (Sagnac interferometer) according to some embodiments of the present invention; and

FIG. 3 is a graph illustrating output transmission probability of a SI (Sagnac interferometer) according to some embodiments of the present invention.

It will be appreciated that for simplicity and clarity of illustration, elements shown in the figures have not necessarily been drawn to scale. For example, the dimensions of some of the elements may be exaggerated relative to other elements for clarity. Further, where considered appropriate, reference numerals may be repeated among the figures to indicate corresponding or analogous elements.

DETAILED DESCRIPTION OF THE PRESENT INVENTION

In the following detailed description, numerous specific details are set forth in order to provide a thorough understanding of the invention. However, it will be understood by those skilled in the art that the present invention may be practiced without these specific details. In other instances, well-known methods, procedures, and components have not been described in detail so as not to obscure the present invention.

The present invention may provide a Sagnac interferometer employing guided matter wave. The interferometer according to embodiments of the present invention may operate with high flux of particles with wide bandwidth, thus providing a clear and evident signal over transient noise. The interferometer according to embodiments of the present invention may provide better rotation sensitivity than prior interferometers by using matter-waves instead of light waves. The interferometer according to embodiments of the present invention may further provide better rotation sensitivity relative to for example, previous matter-wave SIs (Sagnac interferometers) by providing larger effective area of the SI loop, for example, without enlargement of the real area occupied by the loop.

Reference is now made to FIG. 1, which is a schematic illustration of a SI (Sagnac interferometer) 100 according to some embodiments of the present invention. SI 100 may include a loop guide 110, along which matter-waves of particles may propagate in clockwise or counterclockwise direction, for example, in trajectories a or b respectively. SI 100 may include input port i, through which, for example, matter-waves may enter the loop. Input i may be connected to loop 110 by splitter 130, which may split the matter-wave, for example, upon entrance, to two matter-waves propagating in opposite directions, for example, through loop ports α and β. SI 100 may include output port o, through which the particles may exit loop 110. Output port o may be connected to loop 110 by splitter 120, through which the particles may exit to output port o in a certain probability (1−R) or may stay in the loop trajectory in a complimentary probability R. As the probability R is larger, the particles may do more passes through loop 110 before leaving loop 110 through output port o. The paths of the particles through port α or β to splitter 120 may have lengths L_(α) or L_(β), respectively. If L_(α)≠L_(β), there may be a phase difference between the matter-waves in trajectories a and b, which may be proportional to the velocity of the particles in the matter wave. Loop 110 may rotate at a rotation frequency Ω relative to inertial frame 140. If loop 110 rotates, for example, clockwise, particles propagating clockwise in the loop through trajectory a may reach the splitter 120 with an additional negative phase shift, while particles propagating anti clockwise through trajectory b may reach splitter 120 with a positive phase shift. The phase difference between the two trajectories may determine the output amplitude a_(out). Thus, the phase shift between the matter-waves in trajectories a and b may change with the rotation frequency Ω. The rotation velocity Ω may be deduced, therefore, from the output amplitude a_(out). In a general embodiment of an SI, if other origins for velocity (or momentum) dependent phase shifts between the two trajectories exist, such as, for example, when L_(α)≠L_(β), the interference pattern at the output port o may be smeared out and the rotation sensitivity may disappear if the initial beam of particles is a mixture of many velocities.

SI 100 may receive input flux of particles through input port i, for example, with momentum bandwidth Δk. If L_(α)≠L_(β) and Δk is relatively wide, e.g., ΔkδL>>1 (δL=L_(α)−L_(β)), the interfering counter-propagating fluxes of wide range of longitudinal momentums from ports α and β may cancel each other at port o, so that, for example, the dependency of the output at port o on the rotational phase shift φ_(matter) and/or the rotation frequency Ω may be substantially weakened. In this case, the rotational sensitivity of SI 100 may be damaged. Only counter-propagating waves which follow trajectories with substantially the same length may contribute visible dependency in the rotational phase shift in the output. In some embodiments of the present invention, in the case of L_(α)≠L_(β), internal reflections may enable existence of opposite paths with same length.

In case most of the particles follow trajectories with substantially the same length, e.g., the path lengths L_(α) and L_(β) are equal, Δk may be relatively wide without damaging the rotational sensitivity of SI 100. Thus, for example, high flux of particles with wide bandwidth is enabled, thus providing a clear signal which may be evident over transient noise. This may facilitate very sensitive detection of small changes in rotation frequency Ω.

The area A is the effective area enclosed by the trajectory of the particles. As mentioned above, particles may exit loop 110 through splitter 120 in a certain probability (1−R) or may stay in the loop trajectory in a complimentary probability R. As the probability R is larger, the particles may do more passes through loop 110 before leaving loop 110 through output port o. As the particles do more passes, the effective area A is larger, thus, for example, the rotational sensitivity of SI 100 may be greater.

Reference is now made to FIG. 2, which is an exemplary illustration of a SI (Sagnac interferometer) 200 according to some embodiments of the present invention. SI 200 may include a guiding loop 210, BS (beam splitter) 220, BS 230, input port i and output port o. BS 230 may have substantially equal transmission and reflection probabilities, e.g., which may be substantially equal to 0.5. BS 230 may split particle beam incident from input port i into two beams, e.g., reflected beam and transmitted beam, with substantially equal probabilities and, for example, a phase difference of approximately π/2 between them. BS 220 may have reflection probability r² which may be much higher than the transmission probability t², such that r²+t²=1. BS 220 may split a particle beam into two beams, e.g., reflected beam and transmitted beam, wherein, for example, the reflected beam may have much higher probability than the transmitted beam and, for example, the phase difference between them may be of approximately π/2. The transmission and reflection amplitudes may be controllable by, for example, a magnetic tunneling bather and/or any other suitable means for transmission of particle waves between channels with controllable probability, for example, by using magnetic fields. As the reflection probability r² is higher than the transmission probability t², a particle may propagate in loop 210 for a large number of times, thus, for example, enlarging the effective area A and/or the rotational sensitivity of SI 100.

Loop 210 may rotate at a rotation frequency Ω relative to inertial frame 240. As described above with reference to FIG. 1, if loop 210 rotates, for example, clockwise, particles propagating clockwise in the loop may reach the splitter 220 with an additional negative phase shift, while particles propagating anti clockwise through trajectory b may reach splitter 220 with a positive phase shift. The phase difference between the two trajectories may determine the output amplitude a_(out). Thus, for example, the phase shift between the matter-waves in trajectories α and β may change with the rotation frequency Ω. The rotation frequency Ω may be deduced, therefore, from the output amplitude a_(out).

According to some embodiments of the present invention, a particle, for example, an atom, incident from port i may exit at port o through one of four kinds of trajectories: (a) transmission through BS 230, reflection at BS 220 and then transmission through BS 230 again. The total excess phase which may be gained by this path is approximately π; (b) reflection at BS 230, reflection at BS 220 and then reflection at BS 230 again, with substantially no excess phase; (c) transmission at BS 230, transmission at BS 220, propagation in the counterclockwise direction through loop 210 for a certain number of times and then transmission again through BS 220 and then through BS 230, with a total excess phase of approximately 2π and a phase due to Sagnac effect because of the propagation through loop 210; and (d) reflection at BS 230, transmission at BS 220, propagation in clockwise direction in loop 210 for a certain number of times and then transmission again through BS 220 and then through BS 230, with total phase of approximately n and a phase due to Sagnac effect because of the propagation through loop 210. Trajectories (a) and (b) and trajectories (c) and (d) may have substantially the same length.

When loop 210 is not rotating relative to inertial frame 240 the phase in trajectories (a) and (b) may be substantially opposite and the phase of trajectories (e) and (d) may be substantially opposite, thus, for example, the probability to exit at port o may be substantially zero, due to, for example, full destructive interference. However, when loop 210 is rotating relative to inertial frame 240 the particle beams propagating clockwise and counterclockwise through loop 210 may add opposite phase shifts proportional to the rotation frequency Ω and to the number of passes through loop 210. These phase shifts may increase the probability for a positive output of particles at port o, wherein the probability may depend on the rotation frequency Ω. Therefore, for example, the rotation frequency Ω and/or changes in the rotation frequency Ω may be deduced by measurement of output transmission at port o.

In the embodiment of the present invention shown in FIG. 2, a large majority of the particles may follow trajectories with substantially the same length, for example, trajectories (c) and (d) described in detail above. Therefore, it may be possible to use SI 200 with high flux of particles with wide momentum bandwidth, substantially without decreasing the rotational sensitivity of SI 200.

In this embodiments of the present invention, for example, the output transmission probability at port o as a function of the rotational phase φ(Ω), which may be notated P(φ), may be approximately represented by

${\overset{\_}{P}(\varphi)} = {{\frac{t^{2}}{1 + r^{2}}\left\lbrack {1 - \frac{\cos^{2}\varphi}{1 + {4\left( {r/t^{2}} \right)^{2}\sin^{2}\varphi}}} \right\rbrack}.}$

Accordingly, for example, the output transmission probability P(φ) may be substantially zero when the rotational phase (Sagnac phase) φ is substantially zero, e.g., when the rotation frequency Ω is substantially zero. When φ>>t²/r, , the output transmission probability P(φ) may asymptotically approach t²/2r. The thick curve in FIG. 3 is a graph illustrating the output transmission probability P(φ). Other curves in FIG. 3 illustrate output transmission probabilities of SIs according to embodiments of the present invention having imperfections in the guiding potential of the loop, for example, with reflection amplitudes up to 0.25. Thus, the dependency of output transmission probability P(φ) in rotation frequency Ω may be stronger as t/r is smaller.

The sensitivity of the output transmission at port o to changes in the rotation frequency Ω in this embodiment of the present invention, may be defined as the minimum change in the rotation frequency Ω which may generate a noticeable (beyond noise level) change in the output transmission at port o. This minimum change may be represented by

${{\delta\Omega}_{\min} = {\frac{h}{4\pi \; {{mAP}^{\prime}(\varphi)}}\sqrt{\frac{P(\varphi)}{N}}}},$

wherein P′(φ) is the derivative of the output transmission probability with respect to the Sagnac phase φ and N is the total number of incident particles integrated over the time of measurement. Thus, for example, the rotation sensitivity may be maximal when the Sagnac phase φ is substantially zero, e.g., when the rotation frequency Ω is substantially zero. Additionally, for example, the rotation sensitivity may be proportional to the square root of the transmission amplitude t through BS 220.

A Sagnac interferometer according to embodiments of the present invention may provide high rotational sensitivity due to, for example, the ability to propagate particles in loop 210 for a large number of times, thus, for example, enlarging the effective area A and/or the rotational sensitivity of the Sagnac interferometer. Therefore, for example, the actual size of a Sagnac interferometer according to embodiments of the present invention may be very small, for example, having a loop radius of 1 cm. This may enable, for example, implementation of such SI on an atom chip.

The maximum velocity of the atoms in this embodiment may be limited by the magnetic force by which the SI may be bounded to an atom chip. For example, the SI may be bounded to the chip by magnetic field gradients of approximate order of Gauss/μm, which may be generated, for example, by wires on the chip, for example, about 10 μm from the surface of the chip. The centrifugal force mv²/r of the circulating atoms should be below the magnetic force bounding the SI to the chip. For example, the maximum velocity of the atoms may be about 10 m/sec.

As described above with reference to FIG. 2, the time for which the atoms may stay in the SI may depend on the transmission probability t² of BS 220. The transmission amplitude of BS 220 may be, for example, about t=0.035. This transmission amplitude may allow an atom to stay in the SI for about, e.g., 10 seconds, which may enable about 1000-2000 rotations in the guiding loop. For an input flux of, for example, 10⁹ atoms per second, a rotation sensitivity of about 5×10⁻¹² rad·s⁻¹/√{square root over (Hz)} may be obtained.

While certain features of the invention have been illustrated and described herein, many modifications, substitutions, changes, and equivalents will now occur to those of ordinary skill in the art. It is, therefore, to be understood that the appended claims are intended to cover all such modifications and changes as fall within the true spirit of the invention. 

1. An apparatus comprising: a matter-wave guide enclosing an area, wherein a flux of particles may be guided in said matter-wave guide in at least two opposite paths, said apparatus has a rotational velocity relative to an inertial frame of reference; a first beam splitter to split a first beam to at least second and third beams, each of said second and third beams is to be guided in another path of said two opposite paths; and a second beam splitter allowing particles of said second and third beams to exit said matter-wave guide in a first probability and to stay in said matter-wave guide in a second probability.
 2. The apparatus according to claim 1, wherein said first beam splitter allows transmission of particles of said first beam in a third probability and reflection of particles of said first beam in a fourth probability.
 3. The apparatus according to claim 2, wherein said third and fourth probabilities are equal.
 4. The apparatus according to claim 1, wherein said second beam splitter allows transmission of particles of said second and third beams in said first probability and reflection of particles of said second and third beams in said second probability.
 5. The apparatus according to claim 4, wherein said first probability is smaller than said second probability.
 6. The apparatus according to claim 1, wherein said two opposite paths are equal in length.
 7. The apparatus according to claim 1, wherein said second beam splitter allows exit of particles of said second and third beams through an output port of said apparatus.
 8. The apparatus according to claim 7, wherein output transmission probability at said output port is substantially zero when rotation frequency of said matter-wave guide is substantially zero.
 9. The apparatus according to claim 7, wherein dependency of output transmission probability at said output port in rotation frequency of said matter-wave guide is stronger as the ratio between said first probability and said second probability is smaller. 